I'm trying to solve this(Classical Propositional Logic) :
$$ \gamma \ \lor ( \lnot \ \gamma \ \lor (\ \psi \land \ \phi \ )) \, $$
I did this :
- Assuming $$\psi,\phi$$
- Introduction the $$\land$$ so $$(\ \psi \land \ \phi \ )$$
- Now I can use the introduction of $$\lor$$ so $$\gamma \ \lor (\ \psi \land \ \phi \ )$$ 4).And now?
Sorry I my first post, and also I'm new in logical deduction, so please halp me and don't rate me wrong.
P.S I use Tree Proofs
Let $A = (\gamma \ \lor ( \lnot \ \gamma \ \lor (\ \psi \land \ \phi \ ))$. We want to prove $\vdash A$.
Assume $\neg A$ and $\gamma$.
From $\gamma$, by $\lor I$ we have $(\gamma \ \lor ( \lnot \ \gamma \ \lor (\ \psi \land \ \phi \ ))$, then by $\neg E$ we get $\bot$.
Now from $\bot$ using $\neg I$ we can discharge $\gamma$ and deduce $\neg \gamma$.
From $\neg \gamma$ using $\lor I$ we get $\lnot \ \gamma \ \lor (\ \psi \land \ \phi \ )$.
Using $\lor I$ again we can deduce $A$, and from $A$ and $\neg A$, using $\bot$ law, we get $\bot$.
From $\bot$, using $RAA$ we get $A$ and discharge $\neg A$, and the proof is complete.